Problem: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{r^2 - 25}{r + 5}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = r$ $ b = \sqrt{25} = 5$ So we can rewrite the expression as: $t = \dfrac{({r} + {5})({r} {-5})} {r + 5} $ We can divide the numerator and denominator by $(r + 5)$ on condition that $r \neq -5$ Therefore $t = r - 5; r \neq -5$